We investigate compositional iteration of fractional order for transseries.
For any large positive transseries T of exponentiality 0, there is a family
T[s] indexed by real numbers s corresponding to teration of order s.
It is based on Abel's Equation. We also investigate the question of whether
there is a family T[s] all sharing a single support set. A subset of the
transseries of exponentiality 0 is divided into three classes ("shallow",
"moderate" and "deep") with different properties related to fractional
iteration.Comment: 17 pages; 1 figur